No, all satellites do not orbit Earth at the same altitude. An good overview of this can be found on http://www.idirect.net/Company/Satellite-Basics/How-Satellite-Works.aspx. This overview reviews Low Earth Orbit, Medium Earth Orbit and Geostationary Orbits.
Saturn has at least two natural satellites (moons) sharing the same orbit. There may be three altogether, but I only know of two.
Gravity
It has to do with the formation of the solar system. Since all of the matter that condensed into what makes the solar system bodies was spinning in the same direction while coming together, it would have taken a drastic event to cause one of the bodies to orbit in a different direction.
Obviously. Since they move in an ellipse around the Earth (or other central body), they change direction all the time. The only way NOT to change direction would be to move in a straight line; satellites don't do that.
Do all planets and satellites lie on the same gravitational layer?
Saturn has at least two natural satellites (moons) sharing the same orbit. There may be three altogether, but I only know of two.
False. While most comets orbit the Sun in the same direction as the planets (counterclockwise when viewed from above Earth's North Pole), there are exceptions. Some comets have retrograde orbits, meaning they orbit in the opposite direction of the planets.
Yes, the satellites orbit in a geosynchronous orbit, as with most all communications satellites. (Some exceptions are satellites such as the global positioning satellites.)
no
Satellites are placed into different orbits to serve various purposes like communication, weather monitoring, and scientific research. Each orbit has specific requirements, and therefore satellites need to move at different speeds and directions to fulfill their designated tasks. Collisions can occur due to the vast number of satellites in space and the complexities of orbital mechanics, making it impractical to have all satellites move at the same speed and direction.
No. That only applies to low Earth orbit. Geostationary satellites orbit beyond it.
Gravity
You would see the satellite in the same direction all the time. The satellite is supposed to move around the Earth at the same speed as Earth's rotation.
No, not all satellites are moons. Satellites are objects that orbit a planet or other celestial body, while moons specifically refer to natural satellites that orbit planets. Artificial satellites, such as those used for communication or scientific purposes, are not considered moons.
All satellites pass the equator. Twice every orbit.
Yes. All the planets orbit the sun in the same direction. Although, they do not orbit at the same speed, they all orbit in the same direction. The terrestrials (Mercury, Venus, Earth, Mars) orbit at a much faster rate due to their distance from the sun. The Jovians (Jupiter, Saturn, Uranus, Neptune) circle the sun much slower because they are much further away.
All satellites follow an elliptical orbit - they are darn close to circular, but even a circle is an ellipse.